Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [approximation]

For questions that involve concrete approximations, such as finding an approximate value of a number with some precision. For questions that belong to the mathematical area of Approximation Theory, use (approximation-theory).

Approximations are representations of numbers that aren't exact, for example $\sqrt{2}\approx 1.41$. Such representations may be obtained using differentials (more generally, Taylor's formula), linear interpolation, etc.

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The difference between a function and its linear approximations

Suppose that we have a nonlinear $f(x)$ which may be convex/nonconvex. We want to linearize this function in $N$ points over an arbitrary interval $[a,b]$ in such a way that the resultant $N$ linear approximations gives the least deviation from the…
Behnam Alizadeh
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How did the approximation $\left(1+\frac{h}{R}\right)^{-2} \approx 1-\frac{2h}{R}$ come about?

In physics there is a formula calculate the variation in the acceleration due to gravity at a height, it is given by: $$g=g_0\left(1+\frac{h}{R}\right)^{-2}$$ But it is approximated as $$g=g_0\left(1-\frac{2h}{R}\right)$$ Some how the terms in the…
Glowingbluejuicebox
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Approximation of a sum within an exponential

There is a well-known equation in international economics, called the gravity equation. This equation expresses imports from a country as a function of importer size, exporter size, and distance. In…
ChinG
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A question about q-Pochhammer symbol

I am trying to approximate the expression $2^{n^2} (2^{-n};2)_n$, where $(2^{-n};2)_n$ the q-Pochhammer symbol. Actually, I need a formula for the asymptotics of the Pochhammer symbol and $2^{n^2}$ as $n \to \infty$. One such formula is provided…
George
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Need some help understanding notation for composite gauss quadrature formula

Reading through some notes on 2-point gauss quadrature, I came across the following general formula. I'm currently doing an assignment with 3-point quadrature, and have gotten to a similar formula, but now am trying to work out how to evaluate…
Stephen Young
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With a few data points can a generate a close equation to meet them?

I have 1x = -40, 2x = -41 , 3x = -54 and getting a few more. Can I generate a equation for a graph that gets close to this? I was trying to get Wolfram Alpha to do this. EDIT: Sorry for lack of clarity I am hand measuring signal to distance in…
Seth Hikari
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How to get the value of the root?

I have this statement: If $\frac{\sqrt{7}}{\sqrt{5}+\sqrt{3}} \approx \frac{2}{3}$, Which of the following values are the closest to $\sqrt{21}$ ? A) 68/15 B) 14/3 C) 19/4 D) 55/12 E) 9/2 My development was: $\frac{\sqrt{7}}{\sqrt{5}+\sqrt{3}} …
ESCM
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tanh implementations for FPGA neural nets

In trying to put a neural network on my FPGA, I am running into the problem of describing my activation function. On my computer, I typically use the well-known tanh, so I am inclined to stick to my choice on the board too. However, I can't seem to…
Swat Kat
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Linear Approximation Question

So I was given this question: Use the linear approximation to $f(x, y) = \sqrt{\frac {x+5} {y+2}}$ at $(2, 3)$ to estimate $\sqrt \frac {7.1} {4.9}$. I was able to get the actual value by just doing the $\sqrt{7.1 / 4.9}$ which was $1.203735$.…
J.F.
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How to determine significant digits in tolerance of averaged measurements

Consider the scenario where you have a measurement system which records at a resolution of 0.001 but has an accuracy of +/- 0.01. Then, with the following measurement: 8.925 I would report this measurement to three significant figures because the…
Q''
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Can I generalise $x_{n+1}$ in this case?

I have the following algorithm for producing rational approximations of $\sqrt x$. We take an initial $x_0$ and apply the following…
Rhys Hughes
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Name and derivation of the approximation $\frac{1+x}{1+y} - 1 \approx x-y$?

I am wondering if there is a name and way to derive the following approximation: $$\frac{1+x}{1+y} - 1 \approx x-y$$ I'm essentially interested in how to refer to this.
laszlopanaflex
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Big O for Functions Approaching 0

$$f(x) = \text{the Taylor series approximation for } \sin(x)$$ $$f_2 (x) = x$$ where $f_2(x)$ is an approximation for $f(x)$ as x is the first term of $f(x)$. Then: $$g(x) = f(x) - f_2(x)$$ where the first term of $g(x)$ is $-(x^3)/6$. What is…
StopReadingThisUsername
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Any formula that polynomial approximation is not possible (e.g. Taylor expansion)?

Polynomial approximation such as Taylor expansion can deal with many popular functions (sin function, cosine function, etc). I'm curious what kind of functions cannot be approximated by polynomial approximations?
mallea
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